On spectral and orbital stability for the Klein--Gordon equation coupled to an anharmonic oscillator

TL;DR

This paper characterizes the spectral and orbital stability of solitary waves in a Klein-Gordon equation coupled with an anharmonic oscillator, providing a detailed spectral analysis of the linearized operator.

Contribution

It offers explicit criteria for stability and a comprehensive spectral analysis of the linearization at solitary waves in this coupled system.

Findings

Explicit stability conditions derived

Complete spectral analysis of the linearized operator

Characterization of solitary wave stability in the coupled system

Abstract

We obtain explicit characterization of spectral and orbital stability of solitary wave solutions to the $\mathbf{U}(1)$-invariant Klein--Gordon equation in one spatial dimension coupled to an anharmonic oscillator. We also give the complete analysis of the spectrum of the linearization at a solitary wave.